Statistics of cycles: how loopy is your network?

We study the distribution of cycles of length h in large networks (of size N Gt 1) and find it to be an excellent ergodic estimator, even in the extreme inhomogeneous case of scale-free networks. The distribution is sharply peaked around a characteristic cycle length, h_lowast ~ N^α. Our results suggest that h_lowast and the exponent α might usefully characterize broad families of networks. In addition to an exact counting of cycles in hierarchical nets, we present a Monte Carlo sampling algorithm for approximately locating h_lowast and reliably determining α. Our empirical results indicate that for small random scale-free nets of degree exponent λ, α = 1/(λ - 1), and α grows as the nets become larger.